Vidya Venkateswaran

Supercharacters, symmetric functions in noncommuting variables, and related Hopf algebras

We identify two seemingly disparate structures: supercharacters, a useful way of doing Fourier analysis on the group of unipotent uppertriangular matrices with coefficients in a finite field, and the ring of symmetric functions in noncommuting variables. Each is a Hopf algebra and the two are isomorphic as such. This allows developments in each to be transferred. The identification suggests a rich class of examples for the emerging field of combinatorial Hopf algebras.

On the expansion of certain vector-valued characters of

Macdonald polynomials are an important class of symmetric functions, with connections to many different fields. Etingof and Kirillov showed an intimate connection between these functions and representation theory: they proved that Macdonald polynomials arise as (suitably normalized) vector-valued characters of irreducible representations of quantum groups. In this paper, we provide a branching rule for these characters. The coefficients are expressed in terms of skew Macdonald polynomials with plethystic substitutions. We use our branching rule to give an expansion of the characters with respect to the Gelfand-Tsetlin basis. Finally, we study in detail the

U_q (\mathfrak{gl}_n)

q=0

with respect to the Gelfand–Tsetlin basis

case, where the coefficients factor nicely, and have an interpretation in terms of certain

Signature Characters of Highest-Weight Representations of \(U_{q}(\mathfrak {gl}_{n})\)

p

Restricting Supercharacters of the Finite Group of Unipotent Uppertriangular Matrices

-adic counts.

Vanishing integrals for Hall–Littlewood polynomials

We consider Uq(𝔤𝔩n)

Symmetric and nonsymmetric Hall-Littlewood polynomials of type BC

U_{q}(\mathfrak {gl}_{n})

Symmetric and nonsymmetric Koornwinder polynomials in the q → 0 limit

, the quantum group of type A for |q| = 1, q generic. We provide formulas for signature characters of irreducible finite-dimensional highest weight modules and Verma modules. In both cases, the technique involves combinatorics of the Gelfand-Tsetlin bases. As an application, we obtain information about unitarity of finite-dimensional irreducible representations for arbitrary q: we classify the continuous spectrum of the unitarity locus. We also recover some known results in the classical limit q→1

A new class of multiset Wilf equivalent pairs

q \rightarrow 1